As millions marched in cities across the US in a resounding show of defiance against the inauguration of Donald Trump, protesters vowed that the fight had just begun.

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Women’s Marches swept from the canyons of New York to the National Mall in Washington, one day after Trump was sworn in as the nation’s 45th president.

The demonstrations were an opening salvo against a president whose divisive words on the campaign trail – against women, immigrants and Muslims – will not soon be forgotten by a brewing resistance movement.

“This is only the beginning,” Evvie Harmon, global coordinator for the Women’s March, told the Guardian. “We are not going away.

“This is a mass mobilization and we are going to take this network of people and we are going to get them to lobby their members of Congress, call their governors … it’s going on from here.”

Harmon said organisers’ initial estimates indicated that the march in Washington exceeded a million participants while the movement attracted at least 3 million men and women worldwide.

I was hopeless when he won, but now I’m very hopeful. America is not about inequality or racism

Yaye Diop, protester

The huge turnout in the nation’s capital – some inbound flights in recent days were filled almost entirely with female marchers – prompted the event to spill well beyond the official route and clog the city’s main arteries.

Protesters erupted in cheers and chants as they marched, as cars passing by honked in approval.

“I think it’s really powerful to walk,” said one marcher, Jenny Moyrila, who carried a neon-pink sign that read “Wanna Be Starting Somethin’”.

“But then you have to find people hopefully to connect with, to keep on speaking about the issues that matter.”

Moyrila, a librarian on a college campus, said she hoped to find a way to get more students involved in politics. Many others expressed a desire to become more engaged in the political process, galvanized by Trump’s unexpected victory in November’s presidential election.

Khalisa Jacobs and Yaye Diop said they were inspired to continue fighting.

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“There’s not a marginalized group of people that he – I will not say his name for the next four years – hasn’t come against,” Jacobs said. “So people need to take their specific issues and focus on action items that are going to protect their rights.”

Jacobs, who said she intended to attend future protests, said her focus would be equal rights and criminal justice reform. Diop, who emigrated to the US from Senegal, shared a similar sentiment.

“I was hopeless when he won, but now I’m very hopeful,” she said. “America is not about inequality or racism. We are all together – no matter where you come from or what your accent is.”

A man and his daughter participate in the Women’s March on Washington, the day after Trump’s inauguration. Photograph: Lucy Nicholson/Reuters

In several cities, crowds became too good for formal marches to be held safely. In Washington, celebrities and feminist leaders from Scarlett Johansson to Gloria Steinem delivered impassioned speeches just opposite the stage where Trump took the oath of office.

Seas of women and men wearing pink knit-hats engulfed entire blocks of downtown Chicago, Los Angeles and Boston, in a show of advocacy not just for women’s rights but for racial and social justice, LGBT rights and the environment.

We are here. We will not be silent. We will not play dead. We will fight for what we believe in!

“We can whimper, we can whine – or we can fight back!” Warren said, to a crowd in which some waved signs quoting her memorable warning to Trump during the 2016 campaign: that “nasty women fight back”.

She added: “We come here to stand shoulder to shoulder to make clear: We are here. We will not be silent. We will not play dead. We will fight for what we believe in!”

The scenes in Washington offered a sharp contrast to Trump’s inauguration on Friday, when crowd sizes paled in comparison to those which turned out to see Barack Obama’s inaugurations in 2009 and 2013.

Trump used a speech at the CIA on Saturday afternoon to attack the media for, he claimed, distorting its reporting of his crowd sizes. Notably, he did not mention the women’s marches, which in Washington drew more people to the streets than his inaugural ceremony and parade.

In Washington Amita Shukla, 18 and from New York City, was attending her first rally. She was hooked. She planned to become more involved on the issue of women’s reproductive rights, she said, adding: “Step one, march of the nasty women. Step two, grab them by the policy.”

Hours earlier, she had grabbed a free sign that read: “A Woman’s Place is in the Struggle.”

Paulette Gerkovich said she had worked in support of feminist and diversity inclusion for 30 years. She was the first in her family to finish high school but was told she could not go to college. She eventually earned a PhD.

“I have to be here for other women and girls who are told they can’t do something,” Gerkovich said. “They can, and I want to help them understand that.”

I just do not believe that government should be able to tell you what you can and cannot do with your body

Danielle Watson Murray

She posed for a photo outside Trump’s Washington hotel with her husband, James Miller, who held a sign that read “I like to grab my wife by her PhD”, a reference to a leaked 2005 tape in which Trump boasted about groping women without their consent.

Danielle Watson Murray also attended the Washington rally with her husband, who held a sign that read “We’re Stronger Together”. She floated the idea of volunteering for Planned Parenthood, which Republicans in Congress are moving to defund.

“I just do not believe that government should be able to tell you what you can and cannot do with your body,” Murray said. “I was raised in a very Christian household, but I do believe that we should have that right.

“I should be more involved, period.”

Rafaela de la Huerta came from Westchester, New York, with three friends dressed as beauty queens from an “alternative Miss Universe”. Wearing a “Miss Resistance” sash, De la Huerta said the march was symbolic of a new movement.